What is the physics meaning of the "trace" (Tr) and how we can calculate it?

Question

What is the physics meaning of the "trace" (Tr) and how we can calculate it?

$$ Z=tr\left \{ e^{\left ( -H \right )} \right \} $$ Where Z is the partition function .

Answer 1

The trace of a matrix is simply the sum of diagonal entries. In operator theory, the matrix element of the operator given by $$\Omega_{ij}=\langle i|\Omega|j\rangle $$ So the trace would look like $$\text{Tr}(\Omega)=\sum_i\langle i|\Omega|i\rangle$$ In the present case $$\text{Tr}(e^{-H})=\sum_n\langle n|e^{-H}|n\rangle =\sum_ne^{-E_n}$$ Note that $$H|n\rangle =E_n|n\rangle $$

Answer 2

Fix a vector space $V$ over a field (say ${\mathbb R}$ or ${\mathbb C}$).

1. There is a natural map $$tr:V^*\otimes V\rightarrow k$$ given by $$tr(f\otimes v) = f(v)$$

2. There is a natural injection $$\phi:V^*\otimes V\rightarrow Hom(V,V)$$ given by $$\phi(f\otimes v)(w)=f(w)v$$

3. Let $S\subset Hom(V,V)$ be the image of $\phi$. Check that $S$ consists of all $f:V\rightarrow V$ with finite-dimensional range. (So in particular if $V$ is finite dimensional, then $S$ is all of $Hom(V,V)$.)

4. Given $\alpha\in S$, define the trace of $\alpha$ to be $$trace(\alpha)=tr(\phi^{-1}(\alpha))$$

5. If you chase through the definitions, you'll find that if $\alpha$ is represented by a matrix, then $trace(\alpha)$ is the sum of the diagonal elements of that matrix. The advantage of checking this is that it will help make sure you understand the definitions. The disadvantage of checking this is that it might lead you to start thinking of the trace as just a sum of diagonal elements, which pretty much destroys all the intuition.

6. You can think of the trace of $f$ (when it is defined) as trying to measure "the dimension of the subspace onto which $f$ projects". Of course for most $f$ there is no such subspace, so the trace does the best it can: You can always write $f$ as a linear combination of projectors, and the trace of $f$ is the corresponding linear combination of dimensions.
   
   More precisely: Write $f=\sum\alpha_if_i$ where the $\alpha_i$ are scalars and $f_i$ is projection onto a subspace $V_i$ of dimension $d_i$. Then the trace of $f$ is $\sum \alpha_id_i$.

7. For another sort of intuition, you can compute that $tr(f)={d\over dt}(\Delta(e^{tf}))$ where $\Delta$ is the determinant.